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Using concatenated quantum codes for universal fault-tolerant quantum gates

Tomas Jochym-O'Connor, Raymond Laflamme

arXiv:1309.3310v1quant-ph

TL;DR

Universal fault-tolerant quantum computation must control error propagation, while standard approaches often require special ancillary-state preparation. This paper proposes combining two codes to obtain universal gates without state distillation, and illustrates the approach with Steane and Reed-Muller codes while trading full code distance for fault-tolerant gates.

Abstract

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We propose a method for universal fault-tolerant quantum computation using concatenated quantum error correcting codes. Namely, other than computational basis state preparation as required by the DiVincenzo criteria [1], our scheme requires no special ancillary state preparation to achieve universality, as opposed to schemes such as magic state distillation. The concatenation scheme exploits the transversal properties of two different codes, combining them to provide a means to protect against low-weight arbitrary errors. We give the required properties of the error correcting codes to ensure universal fault-tolerance and discuss a particular example using the 7-qubit Steane and 15-qubit Reed-Muller codes. We believe that optimizing the codes used in such a scheme could provide a useful alternative to state distillation schemes that exhibit high overhead costs.

INTRODUCTION

Fault-tolerant quantum computation must control error propagation, yet universal gate sets typically require special quantum states. This work proposes concatenating two codes to obtain universal fault-tolerant gates as an alternative to state distillation, focusing on arbitrary single-qubit errors.

INTRODUCTION

  • Fault-tolerant computation must control errors so they do not propagate through coupled systems, while universal computation typically requires special quantum-state preparation.Noise thresholds identify levels at which scalable computation is achievable without exponential resource overhead.Source: INTRODUCTION · p. 1
  • The proposed concatenated-code approach seeks universal fault-tolerant gates without relying on state distillation, focusing on arbitrary single-qubit errors.The authors also briefly describe how the scheme could be generalized to correct t errors.Source: INTRODUCTION · p. 1
  • Magic state distillation promotes transversal Clifford operations to universal computation, but its preparation overhead can reach 90% of qubits in some architectures.The authors identify reducing this overhead as motivation for their different approach.Source: INTRODUCTION · p. 1

PRELIMINARIES

Universal gate sets provide a finite collection of operations that can efficiently approximate arbitrary unitaries. Because no quantum error-correcting code has a universal set of transversal gates, other fault-tolerant implementations are needed.

PRELIMINARIES

  • A universal gate set efficiently approximates any unitary transformation to a target fidelity using a number of gates logarithmic in inverse fidelity, up to a constant power.For multiple qubits, universal single-qubit gates together with entangling gates form a universal set; the example focuses on {H, T, CNOT}.Source: PRELIMINARIES · p. 2
  • Transversal gates limit each logical gate's interaction to at most one underlying qubit location in the single-error setting considered.The text defines error weight as the number of non-identity locations.Source: PRELIMINARIES · p. 2
  • No quantum error-correcting code contains a universal set of transversal gates, motivating alternative methods for fault-tolerant quantum logic.The authors note that handling errors for a finite gate set is more tractable than correcting faults for arbitrary unitary gates.Source: PRELIMINARIES · p. 2

CONCATENATED QEC

The scheme uses two concatenated codes so their properties can compensate for one another. It requires both codes to have distance at least three and imposes conditions on logical gates and recovery operations to support universal fault-tolerant logic.

CONCATENATED QEC

  • The proposed method uses concatenated quantum error correction to achieve universal fault-tolerant gates by exploiting properties of each code to protect against disadvantages of the other.The logical gate on C1 is composed of multiple logical gates on C2 codeblocks.Source: CONCATENATED QEC · p. 2, Figure 1 · p. 2
  • Both C1 and C2 must have distance at least three, and the scheme treats gates as transversal when they are 1-transversal in the single-qubit-error setting.The information is first encoded in C1, whose constituent qubits are then encoded in C2.Source: CONCATENATED QEC · p. 2
  • Every C1 logical gate that is non-transversal must be implementable using gates transversal in C2, while recovery operations on both codes must be globally transversal.Source: CONCATENATED QEC · p. 2

Logical Gates

Concatenation manages error propagation by requiring non-transversal gates at one code level to be implemented transversally at the other. This keeps faults sufficiently sparse or confined to one codeblock for correction.

Logical Gates

  • With sparse errors, propagation from one physical fault to multiple faults need not become a logical fault after the second encoding.Source: Logical Gates · p. 3
  • A non-transversal gate in C1 can propagate a single fault across logical qubits, so its constituent gates must be transversal in C2 to limit each fault to at most one physical error per encoded block.This preserves the possibility of correcting those errors.Source: Logical Gates · pp. 2–3
  • If a C2 gate used in a C1 logical gate is non-transversal, that C1 logical gate must be transversal so a C2-block fault remains confined to that block.C1 can correct arbitrary logical faults on one of its encoded logical qubits.Source: Logical Gates · p. 3

Quantum Error Correction

The scheme uses different correction procedures depending on whether a gate is transversal at the C1 or C2 level. In the Steane–Reed-Muller example, transversal T construction and transversal-at-C1 H construction protect against single physical faults, while sacrificing the larger distance of straight concatenation.

Quantum Error Correction

  • After a nontransversal C1 gate built from transversal C2 gates, transversal syndrome measurement and correction on each C2 block contain propagated errors.C1-level correction is unnecessary after this gate type when only one fault occurs.Source: Quantum Error Correction · p. 3

EXAMPLE: A [[105,1,3]] QUANTUM ERROR CORRECTING CODE

  • The example concatenates seven-qubit Steane C1 with fifteen-qubit Reed-Muller C2, yielding seven 15-qubit blocks encoding one qubit.Pauli, S, and CNOT operations and syndrome measurements are transversal across the full concatenated code.Source: EXAMPLE: A [[105,1,3]] QUANTUM ERROR… · pp. 3–4
  • The logical T combines transversal CNOT15 and T15 operations across C2 blocks, so one fault spreads to at most three blocks with one error per block.Individual C2-block syndrome measurements then correct the errors.Source: Figure 2 · p. 4, EXAMPLE: A [[105,1,3]] QUANTUM ERROR… · p. 4
  • For logical H, each C2 block receives a nontransversal H15; a single resulting C2 logical fault is detectable by subsequent global C1 correction.The C1 correction measures six Steane stabilizers, whose maximum weight is 32; their measurement is the most expensive correction step.Source: Figure 3 · p. 4, EXAMPLE: A [[105,1,3]] QUANTUM ERROR… · p. 4
  • The [[105,1,3]] construction sacrifices straight concatenation's [[105,1,9]] distance to protect against arbitrary single-qubit faults during logical gates.The paper notes that propagation during logical gates can turn multiple errors into logical faults.Source: EXAMPLE: A [[105,1,3]] QUANTUM ERROR… · pp. 4–5

GENERALIZING TO QECC CORRECTING AGAINST t ERRORS

The scheme can be generalized to codes correcting arbitrary weight-t errors. This requires corresponding t-transversal gate compositions and 1-transversal error-correction operations.

GENERALIZING TO QECC CORRECTING AGAINST t ERRORS

  • For codes correcting arbitrary weight-t errors, a gate that is not t-transversal in C1 must be composed from gates t-transversal in C2.The error-correction operations must also be 1-transversal to avoid propagating faults into multiple errors at the next computation stage.Source: GENERALIZING TO QECC CORRECTING… · p. 5
  • The generalized scheme requires 1-transversal error correction so faults during correction do not propagate into multiple errors at the next computation stage.Source: GENERALIZING TO QECC CORRECTING… · p. 5
  • The code properties for universal gate sets must be modified when extending the scheme from single-error correction to arbitrary weight-t errors.Source: GENERALIZING TO QECC CORRECTING… · p. 5

CONCLUSION

The paper proposes universal fault-tolerance by concatenating codes with complementary transversal properties. Its Steane–Reed-Muller example limits error propagation, while rigorous thresholds and performance with other codes remain open.

CONCLUSION

  • The concatenation uses transversal properties of two codes to keep errors sparse across blocks or confined within a single block.Source: CONCLUSION · p. 5
  • The paper demonstrates the scheme with the seven-qubit Steane and fifteen-qubit Reed-Muller codes.Source: CONCLUSION · p. 5
  • Rigorous thresholds and performance with other quantum error-correcting codes remain open questions.Source: CONCLUSION · p. 5
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